Find the number of different selections of 4 letters from the 9 letters of the word HAPPINESS which contain no P's and either one or two S's.

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Find the number of different selections of 4 letters from the 9 letters of the word HAPPINESS which contain no Ps and either one or two Ss

To find the number of different selections of 4 letters from the 9 letters of the word HAPPINESS which contain no P's and either one or two S's, we can use combinations and counting principles.

Step 1: Count the total number of letters in the word HAPPINESS.

HAPPINESS has a total of 9 letters.

Step 2: Count the total number of P's in the word HAPPINESS.

There are a total of 2 P's in the word HAPPINESS.

Step 3: Count the total number of S's in the word HAPPINESS.

There are a total of 2 S's in the word HAPPINESS.

Step 4: Determine the number of selections with no P's and either one or two S's.

To do this, we need to consider two cases:

Case 1: Selecting one S

In this case, we need to choose 3 more letters from the remaining 7 letters (since we have already selected one S). We need to ensure that none of these selections contain any P's.

The number of ways to select 3 letters from the remaining 7 letters without any P's is given by 7C3.

Case 2: Selecting two S's

In this case, we need to choose 2 more letters from the remaining 7 letters (since we have already selected two S's). Again, we need to ensure that none of these selections contain any P's.

The number of ways to select 2 letters from the remaining 7 letters without any P's is given by 7C2.

Step 5: Calculate the total number of selections.

To find the total number of selections, we add the results from Case 1 and Case 2:

Total number of selections = 7C3 + 7C2

Calculating the combinations using the formula nCr = n! / (r!(n - r)!), where n is the total number of items, and r is the number of selected items, we have:

Total number of selections = (7! / (3! * (7-3)!)) + (7! / (2! * (7-2)!))

Simplifying the equation:

Total number of selections = (7! / (3! * 4!)) + (7! / (2! * 5!))

Total number of selections = (7 * 6 * 5 * 4!) / (3 * 2 * 1) + (7 * 6 * 5!) / (2 * 1)

Total number of selections = 35 + 210

Total number of selections = 245

Therefore, there are 245 different selections of 4 letters from the word HAPPINESS which contain no P's and either one or two S's.